Ratios and proportion in workplace problems: guide and free practice

Before you begin

Be comfortable multiplying, dividing and converting units. The numerical reasoning introduction provides a starting point. The existing general ratio and proportion topic remains available; this lesson focuses on workplace quantities, resource planning and changing mixes.

Read a ratio in order

A ratio compares quantities. Support staff to analysts of 3:5 means three support staff for every five analysts. Reversing the labels reverses the ratio. Both sides can be multiplied or divided by the same positive number without changing the comparison. Convert quantities measuring the same thing to common units before simplifying: 40 minutes to two hours is 40:120, or 1:3.

Part to part is different from part to whole

If support to analysts is 3:5, there are eight parts altogether. The support share is 3/8, not 3/5. Write the labels next to each number before calculating a percentage. A ratio alone gives relative quantities, not their absolute sizes: 3:5 could describe eight staff, sixteen staff or a much larger group.

Worked example: share a total

A fictional project budget of £5,400 is divided between research and delivery in the ratio 4:5. Nine parts represent £5,400, so one part is £600. Research receives £2,400 and delivery £3,000. Check both the sum and the ratio: the shares add to £5,400, and 2,400:3,000 simplifies to 4:5.

When the difference is given

Two teams are in the ratio 3:7 and differ by twenty people. The difference is four parts, so one part is five people. The teams have fifteen and thirty-five people. Dividing the difference by all ten parts would confuse a difference with a total.

Connect ratios through their common quantity

If A:B is 2:3 and B:C is 4:7, make B equal in both comparisons. Multiply the first ratio by four to get 8:12, and the second by three to get 12:21. The combined ratio is A:B:C = 8:12:21. The original B values of three and four are relative parts, not contradictory counts.

Changing a ratio needs quantities

Suppose two stocks are 2k and 5k. Adding nine items to the first changes the ratio to 1:2. Write (2k+9)/(5k)=1/2. This gives 4k+18=5k, so k=18. The original stocks were 36 and 90. Adding the same number to both sides generally changes their ratio; multiplying both by the same factor preserves it.

Direct and inverse proportion

Direct proportion keeps a quotient constant: if every copy takes the same time, twice as many copies require twice as much time. Inverse proportion keeps a product constant: for one fixed job, doubling equally productive machines halves completion time only if the machines can work together without losses or other limits. The question must supply those assumptions; they are not universal facts about real workplaces.

Worked example: more than one resource

Four identical machines produce 1,440 parts in six operating hours. Their rate is 1,440/(4×6)=60 parts per machine-hour. Five such machines in eight operating hours produce 5×8×60=2,400 parts. Keep machine count and operating time separate rather than scaling only one of them.

Mixtures and combined groups

Do not add the numbers in two ratios when the batches have different sizes. Recover each component's amount from its own batch total, then combine matching components. A ten-litre batch in the ratio 1:4 has two litres of its first component. A separate twenty-litre batch in the ratio 3:2 has twelve litres. Their first-component total is fourteen litres out of thirty, not the share implied by simply adding 1+3 and 4+2.

Common mistakes

• Reversing the requested order, such as analysts to support instead of support to analysts.
• Dividing a total by just one side instead of the sum of all parts.
• Mixing hours with minutes or centimetres with kilometres.
• Treating a transfer as an addition: moving staff changes both groups but preserves their total.
• Using direct proportion for a fixed-work problem with inverse completion times.
• Assuming a ratio determines an absolute count when no total, difference or other constraint is supplied.
• Rounding item counts without checking that the original constraints still hold.

Choose your next practice

Foundation covers simplifying, shares, units and direct/inverse scaling. Intermediate combines transfers, linked ratios and changes. Advanced uses equations, multiple resources and mixed batches. Difficulty labels describe teaching steps, not validated ability bands. Continue with weighted averages for unequal groups or table interpretation to choose the right figures from a data source.

Your first practice session

Keep the order of the named quantities. Convert matching quantities to the same units before simplifying. Use constant-rate assumptions only where stated. A calculator is allowed. All business figures are hypothetical.

Try 10 questions with explanations

Choose your practice difficulty

Difficulty labels describe the steps needed in these teaching sets; they are not validated ability bands. Start with foundation practice and move on after you can explain your method.

• Foundation: 10 questions — simplifying, shares, units and direct or inverse scaling.
• Intermediate: 10 questions — Combine linked ratios, staff transfers, changing budgets and direct or inverse scaling in ten original problems.
• Advanced: 10 questions — Use ratio equations, multiple resource factors and combined batches to solve ten original workplace problems.

Continue learning

• Percentage change in workplace data
• Weighted averages and combined rates
• Rates and productivity
• Interpreting workplace tables

About this practice

These are independently written teaching questions. A practice score helps you review your answers; it is not a validated hiring prediction or an official employer score.